We say that a Borchers triple \(({\mathcal {M}}, T, \Omega )\)  is wedge-split if \({\hbox{\rm Ad\,}}T(a)({\mathcal {M}}) \subset {\mathcal {M}}\) 
is split for any \(a\in W_{\mathrm {R}}\) . Note that \(W_{\mathrm {R}}\)  is defined as the open wedge
and split inclusion for lightlike translation is not required.
It is easy to see already from wedge-split property that the intersection \({\mathcal {M}}\cap {\hbox{\rm Ad\,}}T(a)({\mathcal {M}})\) 
for \(a\in W_{\mathrm {R}}\)  is nontrivial.
Let \(\Delta \)  be the modular operator for \({\mathcal {M}}\)  with respect to \(\Omega \) .
One says that an inclusion \({\mathcal {N}}\subset {\mathcal {M}}\)  satisfies modular nuclearity if
the map \({\mathcal {N}}\ni x \longmapsto \Delta ^{\frac{1}{4}}x\Omega \)  is nuclear.
A Borchers triple \(({\mathcal {M}},T,\Omega )\)  is said to satisfy modular nuclearity if
the inclusion \({\hbox{\rm Ad\,}}T({\mathcal {M}}) \subset {\mathcal {M}}\)  has modular nuclearity for any \(a\in W_{\mathrm {R}}\) .
From the above remark, modular nuclearity implies wedge-split property.
The strict locality of the models in [1]} was proved through modular nuclearity.
